Scaling UV map to fit bounds
Select one side (left/right) from 1 diagonal to the other, only borders, and scale it on X axis to 0 (S, X, 0) then move it away


Continue with other sides (up/down-> Y axis)



Once all four sides scaled and moved away, select the other vertices, scale them down, then select the borders again


scale and move them to fit your UV bounds 


Select everything but the border vertices and from the UV menu, click minimize stretch (or Ctrl+V)

use mouse wheel to blend the effect to your liking, when it's ok, hit enter